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Electric/Magnetic Dipole in an Electromagnetic Field: Force, Torque

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Electric/Magnetic Dipole in an Electromagnetic Field: Force, Torque Eur. Phys. J. Plus (2014) 129: 215 THE EUROPEAN DOI 10.1140/epjp/i2014-14215-y PHYSICAL JOURNAL PLUS Regular Article Electric/magnetic dipole in an electromagnetic field: force, torque and energy Alexander Kholmetskii1,2,a, Oleg Missevitch3, and T. Yarman3,4 1 Belarusian State University, Minsk, Belarus 2 Institute for Nuclear Problems, Belarusian State University, Minsk, Belarus 3 Okan University, Akfirat, Istanbul, Turkey 4 Savronik, Eskisehir, Turkey Received: 16 June 2014 Published online: 9 October 2014 – c Societ`a Italiana di Fisica / Springer-Verlag 2014 Abstract. In this paper we collect the relativistic expressions for the force, torque and energy of a small electric/magnetic dipole in an electromagnetic field, which we recently obtained (A.L. Kholmetskii et al., Eur. J. Phys. 33, L7 (2011), Prog. Electromagn. Res. B 45, 83 (2012), Can. J. Phys. 9, 576 (2013)) and consider a number of subtle effects, characterized the behavior of the dipole in an external field, which seem interesting from the practical viewpoint. 1 Introduction The problem of derivation of relativistic expressions for the force and torque experienced by a small electric/magnetic dipole in an external electromagnetic field was the subject of essential interest of researchers at the second half of the 20th century, and continues to attract attention up to date. The essential progress in this area was achieved with the derivation of the Bargmann-Michel-Telegdi (BMT) equation (1), which describes the time evolution of the particle’s spin with the inclusion of Thomas precession [2]. At the same time, the approach used in the derivation of the BMT equation (the introduction of the spin four-vector with the vanishing time component in the proper frame of the particle [1,3]) is convenient to determine the motion of the particle’s spin in its proper frame in terms of the external electromagnetic (EM) field measured in the laboratory. However, we remind that the transformation of the torque components is guided by the corresponding torque four-tensor [4], and one can check that its application does not allow to obtain an explicit analytical expression for the torque in the laboratory frame. At the same time, the derivation of the force and torque expressions, where all quantities are measured in the laboratory, has a principal importance in the consistent investigation of relativistic effects, related to the motion of an electric/magnetic dipole in an EM field. In addition, before the discovery of the hidden momentum of magnetic dipoles [5], its contribution to the force and torque was not taken into account; moreover, it continues to be ignored in some modern publications, including textbooks (e.g., [6]). One should mention that even in the past years, the problem of deriving the correct expression for the force and torque on a compact dipole faced a number of errors made by various authors (e.g., [7–11]). Nevertheless, the correct Lorentz-invariant expression for the force on a moving electric/magnetic dipole, seemingly for the first time, has been achieved in our recent contribution [12], which opened the way for the derivation of the relativistic equations for the description of a torque on a moving dipole [13], and its electromagnetic energy [14]. Now it seems useful from the practical viewpoint to collect these equations altogether in a single paper and to discuss some subtle points, related to their physical meaning, which were not specially commented in the mentioned papers [12–14]. The obtained equations for the force F , torque T and energy E of a small electrically neutral dipole with the proper electric p0 and magnetic m0 dipole moments, moving at velocity ν in the external electric E and magnetic B a e-mail: [email protected] Page 2 of 13 Eur. Phys. J. Plus (2014) 129: 215 fields, are as follows [12–14]: 1 d 1 d F = ∇(p · E)+∇(m · B)+ (p × B) − (m × E), (1) c dt c dt 1 1 T = p × E + m × B + ν × (p × B) − ν × (m × E), (2) c c 1 1 E = −(p · E) − (m · B) − (p × B) · ν + (m × E) · ν, (3) c c where (γ − 1) ν × m p = p − (p · ν)ν + 0 , (4) 0 γν2 0 c (γ − 1) p × ν m = m − (m · ν)ν + 0 (5) 0 γν2 0 c are, respectively, the electric and magnetic dipole moments of a moving dipole in the frame of observation [6,15,16], and γ =(1− ν2/c2)−1/2 is the Lorentz factor. Here we stress that eq. (3) determines the electromagnetic energy of the dipole only, and does not include the fractions of energy absorbed (extracted) in the power supply and in a source of external magnetic field [14]. We point out that the quantities entering into eqs. (1)-(3) are defined in the frame of observation (laboratory frame), and these equations also can be expressed either via the quantities measured in the rest frame of a dipole, or via some combination of these quantities (e.g., the proper electric and magnetic dipole moments, measured in the rest frame of a dipole, and electromagnetic fields measured in the laboratory. As we will see below, the latter combination is convenient for the analysis of the energy of the dipole). In sect. 2 we discuss the physical meaning of eqs. (1)–(3), focusing mainly our attention on the expressions for torque (2) and energy (3) for a moving dipole, insofar as the expression for the force experienced by a dipole has been already discussed in the related publications [9,11,12]. Finally, we conclude in sect. 3. 2 Expressions for the force, torque and energy of a moving dipole: Origin and implications 2.1 Force on a moving electric/magnetic dipole in an electromagnetic field In this subsection we address eq. (1) and remind that it originates from the Lorentz force law, applied to the electrically neutral compact bunch of charges [9] and also includes the force component due to time variation of the hidden momentum of the magnetic dipole (see, e.g. [3,5,17–19]). More specifically, the sum of the terms contaning p (i.e. the first and third terms) describes the Coulomb interaction of the charges of the electric dipole with the electric field, and the interaction of the convective currents of the dipole with the magnetic field. The sum of the terms contaning m (i.e. the second and fourth terms) is responsible for the interaction of the proper (closed) current of the magnetic dipole with the magnetic field, and also includes the hidden-momentum contribution. A real situation might be more complicated, because the moving electric dipole develops the magnetic dipole moment, and the moving magnetic dipole possesses an electric dipole moment, see eqs. (4), (5). However, we skip the detailed analysis of the various terms in eq. (1), which can be found in the above-mentioned publications [9,11, 12], and would like to give answer to the question, which we ask time by time: why the Lorentz force law alone fails to describe the hidden-momentum contribution to the resultant force on a magnetic dipole, and the latter should be exogeneously added to the motional equation for a bunch of charges in an electromagnetic field? The general answer to this question is straightforward: the Lorentz force law is relevant only for point-like charges, whereas an electric/magnetic dipole always represents some finite-size distribution of the charges. If so, an external electromagnetic field should induce some changes in its inner volume (polarization, magnetization, the appearance of mechanical stresses, etc.). The mentioned changes can be classified as some “secondary” effects emerging in the bulk of the dipole due to the external electromagnetic field. It was pointed out for the first time in ref. [5] that such secondary effects can be attributed to the appearance of the momentum component of the magnetic dipole, even if the dipole, as the whole, is at rest in the frame of observation; that is why this component was named as the “hidden” momentum. Moreover, it was further proven in ref. [17] that for a magnetic dipole, resting in an electric field, being created by a static-charge distribution, the hidden momentum of the dipole is exactly equal with the reverse sign to the interactional electromagnetic momentum of this configuration, and the time variation of the hidden momentum (which induces the appearance of force on the dipole) is exactly equal with the reverse sign to time variation of the field momentum. Therefore, the hidden momentum plays an important role in the energy-momentum balance for isolated systems, which include charges and magnetic dipoles. Eur. Phys. J. Plus (2014) 129: 215 Page 3 of 13 Fhidden Fp q m . v -Fp Fin y x Fig. 1. Interaction of a resting charge q with a moving magnetic dipole m. The dipole is moving without friction along the x-axis inside a thin insulating tube (not shown in the figure), and the charge is rigidly attached to the tube. The tube has a single degree of freedom to move along/opposite to the y-axis. To demonstrate the validity of this statement, let us consider the following problem (see fig. 1). Let a small magnetic dipole with the proper magnetic dipole moment m{0, 0,m} move inside a thin isolated tube oriented along the x-axis. Let a point-like charge q be rigidly fixed inside the same tube at the origin of the coordinates. We further asume that at the initial time moment t = 0 the magnetic dipole has the x-coordinate X1, and analyze the implementation of the energy-momentum conservation law, considering another time moment t = T , when the x-coordinate of the dipole becomes equal to X2.
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